Quantization Effect on the Log-Likelihood Ratio ... - Semantic Scholar
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Quantization Effect on the Log-Likelihood Ratio and Its Application to Decentralized Sequential Detection Yan Wang and Yajun Mei
Abstract—It is well known that quantization cannot increase the Kullback–Leibler divergence which can be thought of as the expected value or first moment of the log-likelihood ratio. In this paper, we investigate the quantization effects on the second moment of the log-likelihood ratio. It is shown via the convex domination technique that quantization may result in an increase in the case of the second moment, but the increase is bounded above by . The result is then applied to decentralized sequential detection problems not only to provide simpler sufficient conditions for asymptotic optimality theories in the simplest models, but also to shed new light on more complicated models. In addition, some brief remarks on other higher-order moments of the log-likelihood ratio are also provided.
is distributed according to for or 1. Then the Kullback-Leibler divergence of the quantized observation is
Index Terms—Convex domination, decentralized detection, Kullback-Leibler, log-sum inequality, quantization, quickest change detection, sequential detection.
(1)
I. INTRODUCTION
I
N information theory and statistics, the Kullback-Leibler divergence is a fundamental quantity that characterizes the difference between two probability distributions and of a random variable . Denote by and the densities of and with respect to some common underlying probability measure , then the Kullback-Leibler divergence from to is
where
is the log-likelihood ratio
and means taking the expectation over the distribution . In some applications, we need to deal with the quantized version of that is a function of and often is required to belong to a finite alphabet. Denote by the quantized observation, and let and be the probability distribution and probability mass (or density) function of when Manuscript received August 20, 2012; revised November 20, 2012; accepted November 27, 2012. Date of publication January 01, 2013; date of current version February 26, 2013. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Huaiyu Dai. This work was supported in part by the AFOSR grant FA9550-08-1-0376 and the NSF grants CCF-0830472 and DMS-0954704. The material in this paper was presented at the IEEE International Symposium on Information Theory, Cambridge, MA, USA, July 1–6, 2012. The authors are with the School of Industrial and System Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TSP.2012.2237170
where
is the log-likelihood ratio of
defined by
An important property is that quantization cannot increase the Kullback-Leibler divergence, that is,
with equality if and only if is a sufficient statistic of , see Theorem 4.1 of Kullback and Leibler [5]. This is conis generally less inforsistent with our intuition that mative than the itself. Note that the inequality (1), which will be referred to as Kullback-Leibler’s inequality below, deals with the expected value or first moment of the log-likelihood ratio. In this paper, we extend the Kullback-Leibler inequality (1) to investigate the quantization effects on the second or other higher moments of the log-likelihood ratio. Our research is motivated by the decentralized sequential detection problem where there ; ; are unobservable raw random variables , and where what is actually observed are their . In such a problem, one quantized versions ’s so as wants to design appropriate quantization functions ’s to make the best possible decision (under a to utilize the to maxproper criterion). Intuitively, it is natural to choose imize the Kullback-Leibler divergence of the quantized obser) if the ’s vations (or possibly or for each . See Tsitare independent with density siklis [18] for the characterization of such an optimal quantizer. In order to guarantee that the corresponding statistical procedures are indeed efficient, one needs to verify that the distrisatisfies some standard bution of quantized observations regularity conditions. One of such conditions is that quantized observations have finite variances, or more rigorously speaking, have that the second moments of the log-likelihood ratios a common upper bound over a class of allowable quantization ’s. functions Unfortunately, it can be analytically challenging or intractable to verify the assumptions for quantized observations ’s are known directly, even if the distributions of the to belong to some simple families of distributions, as one may to work with have a large class of quantization functions when finding the optimal one. To overcome such a difficulty, in this paper we show that one could dispense with quantization and only look at the second moment associated with the raw
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data ’s since the boundedness of the unquantized version implies the boundedness of the quantized version regardless of the quantization function . Also see Le Cam and Yang [7] for similar approaches in other contexts. The remainder of the paper is organized as follows. In Section II we extend the Kullback-Leibler’s inequality (1) to the second moment of the log-likelihood ratio. In Section III, our result is applied to the decentralized sequential detection problems, not only to provide simpler sufficient conditions for asymptotic optimality theories in the simplest models, but also to shed new light on more complicated models. In Section IV, we add several remarks, including the quantization effects on higher-order moments of the log-likelihood ratios. II. SECOND-ORDER MOMENTS For the and , define their respective second moments of log-likelihood ratios as
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To do so, taking derivatives of the function leads to
Hence is convex on but is concave on . Therefore, if we consider the following new function if if
,
is a continuous convex function of when . then Moreover, the concavity of on implies that dominates . Now by the definitions of , and , we have
(2) where and are the log-likelihood ratios of and . Our main result of this paper is as follows. Proposition 1: Denote by the set of all possible measurable function . Then we have (3) Proof: Fix a measurable function , and let and be the likelihood ratios. To simplify notation, let denote the conditional expectation with respect to a given value of the quantization observation , then it is easy to see that
Recall that in the proof of the Kullback-Leibler’s inequality (1), one uses heavily the fact that the function is convex when . By Jensen’s inequality,
where the first inequality follows from the fact that , and the second inequality is an application of Jensen’s inequality to the new convex function . Meanwhile, the difference between and turns out to be insignificant. By definition, for all , and thus
where we use the fact that
Combining the above inequalities yields
for any given measurable function . Taking the supremum over all possible functions leads to (3), completing the proof of our theorem. and the Kullback-Leibler’s inequality (1) is proved by taking expectations under on both sides. Unfortunately, the above approach fails for the second moment case since the function is no longer convex (nor concave). Fortunately, this approach can be salvaged by what we call the “convex domination” technique, i.e., by finding a convex function that is larger, but not too much larger, than .
III. DECENTRALIZED SEQUENTIAL DETECTION The problem that motivated us to write the present paper arises from decentralized sequential detection problems. Below we will illustrate the usefulness of Proposition 1 to get around mathematical challenges in decentralized sequential detection problems, thereby providing new insights to the field.
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different formulations of these problems using different network architectures, we refer Braca et al. [2], [3] and the references herein. A. Decentralized Sequential Hypotheses Testing
Fig. 1. A decentralized sensor network.
As discussed in Bajovic et al. [1], distributed/decentralized detection can be broadly divided into several different classes, depending on parallel or consensus network architectures. Here we will only focus on a specific network system using the parallel architecture with a fusion center, see, Veeravalli, Basar and Poor [21], and Veeravalli [17], [19]. Fig. 1 depicts a typical configuration of such a network system. Suppose there are geographically deployed local sensors and a fusion center which makes a final decision. At each time step , each local sensor observes a raw data and sends a quantized message to the fusion center, which makes a final decision when stopping taking observations. Due to constraints on communication bandwidths or requirements of reliability, the local sensors are required to compress the raw data to quantized sensor messages ’s, which all belong to finite alphabets, say . In other words, the fusion center has no direct access to the raw observations and has to make its decisions based on the quantized sensor messages. If necessary, the fusion center can send feedback, , to the local sensors to adaptively adjust the local quantization so as to achieve maximum efficiency. There are many possible useful setups for the decentralized network system with the parallel architectures, and one widely used setup is the system with limited local memory and full feedback, or Case E in Veeravalli, Basar and Poor [21]. Mathematically, in such a system, the quantized sensor message sent from the local sensor to the fusion center at time is
where
(here ) denotes all past sensor messages at the fu-
sion center. In decentralized sequential detection problems, a fusion center policy often consists of selecting a stopping time at which time step it is decided to stop taking observations, and a final decision . The objective is to jointly optimize the sensor quantization functions ’s at the local sensor level and the fusion center policy so as to make an optimal decision in some suitable sense. Below we will discuss the usefulness of Proposition 1 in two different kinds of decentralized sequential detection problems: sequential hypothesis testing and quickest change detection. For
In the simplest model of decentralized (binary) sequential hypotheses testing problems, there are two simple hypotheses and regarding the distributions of the raw observations ’s, and conditional on each of these two hypotheses, the form i.i.d. sequences over time and are independent among different sensors. The objective is to use as few samples as possible to correctly decide which of these two simple hypotheses is true. That is, one wants to balances the trade-off between the average sample size under each hypothesis and the probabilities of making Type I and II errors. Under the Bays formulation, it is assumed that the two hypotheses and have known prior probability, say, for some , let be the cost of falsely rejecting , and each time step costs a positive amount . Then the Bayes risk of a decentralized sequential test with a stopping time is
where the notation is used to emphasize that the cost for each step is . The Bayes formulation of decentralized sequential hypothesis testing problems can be stated as follows. Problem (P1): Minimize the Bayes risk over all possible sensor quantization functions ’s and over all possible policies at the fusion center. The Bayes solution was characterized in Veeravalli, Basar and Poor [21], which showed that the Bayes solution is based on the “stationary Monotone Likelihood Ratio Quantizer (MLRQ)” quantizers. However, the optimal quantizers were “stationary” only in the sense of dynamic programming, i.e., the quantizers used thresholds to quantize observations, but the thresholds had to be updated at each time step, based on the posterior probability of . So, what was stationary was the functional structure of the quantizers. Specifically, for the Bayes procedure, the sensor messages of the -th sensor at time are formed through the following MLRQs based on the posterior probability :
where the
threshold functions satisfy
for all . While the threshold functions can be approximated via numerical experimentations, there is no closed-form formula for ’s, which appeared to be a discontinuous function of , see Fig. 3 and Fig. 4 of Veeravalli, Basar and Poor [21]. Thus, the usefulness of Bays procedure is rather limited, from both theoretical and practical viewpoint. In order to develop a feasible decentralized hypothesis testing procedure, it is natural to ask whether fixed quantizers can be used to construct decentralized sequential tests that are asymptotically optimal when the cost per time step goes
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to 0. Unfortunately, the answer turns out to be negative, see Nguyen, Wainwright and Jordan [12] for a counterexample. The negative answer is not surprising, because each local sensor will have two kinds of optimal fixed quantizers: one maximizes (if is true) and the other maximizes (if is true), due to the asymmetric properties of the Kullback-Leibler divergences. Earlier Tsitsiklis [18] showed that the optimal quantizer that maximizes the local Kullback-Leibler divergence (or ) is a form of MLRQs. Denote them by and respectively. To develop a simple but asymptotically optimal decentralized sequential tests, Mei [10] introduces the concept of “tandem quantizers” where the test procedure is divided into two stages. In the first stage, any reasonable fixed quantizer is used and the network system makes a preliminary decision about which of two hypothesis is likely to be true. Then at the second stage, each local sensor switches to one of two optimal fixed quantizers or , based on whether the preliminary decision chooses the hypothesis or . The main motivation is that in statistics, first-order asymptotically optimal procedures often mainly depend on maximizing the first moment (or mean) of the log-likelihood ratios as long as the second moments of the log-likelihood ratios are finite. In particular, for the two-stage test with tandem quantizers, if the time steps taken at the first stage is large enough to make a reasonable preliminary decision, but relatively small as compared to those at the second stage, then we will asymptotically maximize the first moment of the log-likelihood ratios of quantized observations simultaneously under both and . Thus, one should expect that the two-stage test with tandem quantizers enjoy some nice properties. Indeed, it was shown in Mei [10] that under the sufficient condition
(4) for all , then the two-stage tests with tandem quantizers are first-order asymptotically optimal among all possible decentralized sequential tests that use any sequence of measurable quantizers, stationary or not. While the sufficient condition (4) has been verified in a couple of specific cases such as normal and exponential distributions with binary sensor messages, unfortunately it is unclear whether (4) holds in general or not. This paper is the result of our continued efforts to better understand the sufficient condition (4). By Proposition 1, the asymptotic optimality theory in Mei [10] holds under a simpler and more general sufficient condition, and can be stated as follows. Theorem 1: As long as the second moments of the log-likelihood ratios of raw sensor observations are finite, i.e., and for all , the two-stage tests with tandem quantizers are first-order asymptotically optimal among all possible decentralized sequential tests that use any sequence of measurable quantizers, stationary or not. We also want to add a remark to emphasize the important role of Proposition 1 in the more complicated model of decentralized sequential hypothesis testing problems in which one wants
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to test multiple hypotheses. Proposition 1 essentially guarantees that when developing asymptotically optimal decentralized sequential multi-hypothesis tests, one does not need to worry about the “strange” behavior of non-stationary quantizers (such as infinite variances), and it is paramount to choose local quantizers “smartly” so as to (asymptotically) maximize the first moment (or mean) of the log-likelihood ratios of quantized observations simultaneously under each and every possible hypothesis. This view allows us to extend the two-stage with tandem quantizers to develop asymptotic optimality theories for decentralized sequential multi-hypothesis testing problems, see [22] for the first-order asymptotic theorem and [23] for the second-order asymptotic theorem in certain cases. B. Decentralized Quickest Change Detection In decentralized quickest change detection problems, it is assumed that an event occurs to the network system at some unknown time , and changes the probability measure of the raw data from (with density for observations ) to (with density ). In the simplest model, it is assumed that all pre- and post-change densities, and , are completely specified, and conditional on each hypothesis on the possible change-time or (no change), the observations are independent over time and from sensor to sensor. Ideally, we want to continue taking observations as long as possible if no event occurs but will stop as soon as possible after the event occurs. A fusion center policy in decentralized quickest change detection problems can be simply defined as a stopping time , as its final decision at the stopping time is to claim that a change has occurred at or before time . There are two standard formulations to decentralized quickest change detection problems. The first one is the Bays formulation in which the change-time is assumed to be geometrically distributed, i.e., for , also see Shiryaev [15] and [16], Veeravalli [20]. Under the Bays formulation, it is further assumed that the cost of raising a false alarm is 1 and the cost of taking a post-change observation is per time step, and the problem can be stated as follows (For alternative minimax formulations, see Lai [6], Lorden [8], Moustakides [11], Page [13], Pollak [14], etc.) Problem (P2): Minimize the Bayes risk
over all possible fusion center stopping times and all possible sensor quantization functions ’s. The Bays procedure is characterized in Veeravalli [20], which again showed that the Bays solution is based on the MLRQ quantizers which were “stationary” only in the sense of the functional structure of the MLRQs, as the thresholds had to be updated at each time step, based on the posterior probability of change. Through numerical simulations, Veeravalli [20] found that as compared to Bays procedure, the loss in performance is insignificant if one simply uses fixed MLRQ quantizers whose thresholds are constant over time and optimized off-line to maximize the Kullback-Leibler divergence . This leads to a conjecture in Veeravalli [20] that the optimal
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fixed MLRQs are sufficient to construct asymptotically optimal decentralized change detection schemes over the class of all possible decentralized schemes that use any sequence of measurable quantizers, stationary or not. Unfortunately, it is not easy to prove or disprove Veeravalli’s asymptotic optimality conjecture, partly because the regularity conditions of the quantized observations are needed to do any reasonable asymptotic analysis. Indeed, all existing research has to put some restrictions on the sensor quantization functions, see Crow and Schwartz [4], Tartakovsky and Veeravalli [17]. A different approach was presented in Mei [9], which showed that Veeravalli’s conjecture holds under the following sufficient condition (5) is the second moment of log-likelihood ratio for quanwhere tized observations defined in (2). Proposition 1 significantly broadens the applicability of the asymptotic optimality theorem in Mei [9], and shows that Veeravalli’s conjecture holds under a simpler sufficient condition as stated in the following theorem. Theorem 2: As long as the second-moments of the log-likelihood ratios of the raw observations are finite, i.e.,
the optimal fixed MLRQs are sufficient to construct asymptotically optimal decentralized change detection schemes over the class of all possible decentralized schemes that use any sequence of measurable quantizers, stationary or not. Proof: By Lai [6], the asymptotic optimality conclusion holds under the following sufficient condition:
(6) where at time , i.e.,
and Here
is the probability measure when the change occurs is the likelihood ratio for the quantized data ,
with is probability mass function, i.e.,
.
By using Kolmogorov’s inequality for martingales, Mei [9] shows that (5) implies (6). By Proposition 1, if , then (5) holds and so does (6). Hence, the asymptotic optimality conclusion is proved. Besides simplifying the sufficient condition of asymptotic optimality theory in the simplest model when all pre- and
post-change densities, and , are completely specified for the local sensor observations, Proposition 1 can also shed new lights on a more complicated model of the decentralized quickest change detection problem when the post-change densities ’s are only partially specified. To the best of our knowledge, little research has been done due to its mathematical difficulties and complexities. To illustrate our main idea, let us focus on a concrete simplified example under a minimax formulation of decentralized quickest change detection problems. Assume there is only sensor, and the local raw sensor observations ’s are independent normally distributed with variance 1, with a possible change in the mean from 0 to one of two possible values, , for some known . The quantized sensor mesare binary, and the fusion center sages wants to utilize the ’s to raise an alarm quickly once a change occurs. Under the minimax formulation, when the true post-change mean is , one widely used definition of the detection delay is
where denotes the expectation of raw data when the mean change from 0 to at time . Now there are two kinds of detection delays: one for and the other for . A good decentralized quickest detection scheme should keep both detection delays small. Thus, our problem can be stated as follows. Problem (P3): Find a decentralized quickest detection scheme that nearly (or more rigorously, asymptotically) minimizes both and simultaneously over all possible fusion center stopping times and all possible sensor quantization functions ’s, subject to the false alarm constraint for some pre-specified constant . Here denotes the expectation when there are no changes. is a fixed quantizer over time , say, When for some threshold , the quantized sensor messages are independent Bernoulli random variable, and the fusion center essentially faces the problem of detecting a change on from to or . Such a problem is well-studied in the centralized quickest change detection problem, and one efficient scheme is the 2-CUSUM procedure defined by
where we use to emphasize that the 2-CUSUM procedure depends on the fixed quantizers in the decentralized quickest change problems. Here is the CUSUM statistic at time in the problem of detecting a change from to and is the CUSUM statistic at time in the problem of detecting a to . Specifically, for , the CUSUM change from statistic can be defined recursively as
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TABLE I THE DETECTION DELAYS OF FOUR PROCEDURES
with
, where and depend on the value of . It is natural to ask which is the “optimal” fixed quantizer , or what is the “optimal” fixed threshold for the 2-CUSUM procedure . It turns out the answer depends on what we mean by “optimal.” If we want to minimize , then we should choose to maximize , where and . On the other hand, if we want to minimize , then we should choose to maximize with . Moreover, it is not difficult to show that if we want to minimize , then we should choose to maximize . For instance, when , numerical computation shows that the corresponding three “op, 0.7941, and 0. timal” thresholds are A more challenging question is to develop a single decentralized change detection scheme that is asymptotically optimal and in the sense of asymptotically minimizing both simultaneously. In this case, the sensor message function cannot be fixed, and the thresh’s must be allowed to depend on the past messages olds . But how to choose the ’s? This is an open question so far, and our Proposition 1 provides some new ideas to tackle this problem. By Proposition 1, to develop asymptotically optimal decentralized change detection schemes, we can ignore the second moment and simply investigate how to choose local quantizers so as to (asymptotically) maximize the first moment (or mean) of the log-likelihood ratios of quantized observations simultaneously under all possible post-change hypotheses. Now we have two possible post-change means, and , and two corresponding optimal fixed thresholds, say for detecting and for detecting a a change in the mean is from 0 to change from 0 to . Thus the question can be further reduced to how to adaptively assign or to local sensor message functions so that it can “self-tune” to match the true post-change mean. Note that the idea of two-stage tests with tandem quantizers in decentralized sequential hypothesis testing problems no longer works for decentralize quickest change detection problems, since the change may happens after the first stage and thus the preliminary decision at the first stage does not necessarily reflect the true change. However, the fundamental
ideas can be salvaged: the local sensor message functions can be optimized according to a preliminary decision. To be more concrete, one possible strategy for decentralized quickest change detection can be defined as follows. Suppose that the fusion center uses the 2-CUSUM-type procedure of the form , with the detection statistic designed for monitoring a change from 0 to and the detection statistic for monitoring a change from . Initially one may use the quantizer 0 to at time . For , the quantizer used at time step will depend on the observed values of and . If , then we may feel that the more likely post-change mean will probably be and thus should be chosen as . On the other hand, if , we may think that the post-change mean is likely and thus . In particular, for , and . Now once we determine at time , the local sensor takes an the specific threshold observation , and sends the message to the fusion center, which uses relation (7) to update both and , except that the values of and depend on the specific value of . We call this procedure 2-CUSUM with adaptive thresholds, and denote it by . , we ran a numerical simulaTo see the advantage of tion for , and compared it with the 2-CUSUM procedure with three different fixed thresholds , 0.7941, 0. For these four procedures, the threshold value was first determined from the criterion , and we then use the value to simulate the detection delays and based on Monte Carlo repetitions. The results are reported in Table I. From Table I, all three 2-CUSUM procedures with fixed quantizers have some weakness: The threshold leads to a procedure that has smallest detection delay but the largest detection delay , and the conclusion . While the was reversed for the fixed threshold fixed threshold leads to a procedure that has similar performances on and , the detection delays are much larger than the smallest possible individual delays we can obtain. Meanwhile, the proposed scheme with adaptive thresholds ’s performs well under both post-change is close to its smallest scenarios in the sense that value for both and . All these are consistent with our theoretical conclusions.
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IV. REMARKS 1) The discrete version of the Kullback-Leibler’s inequality (1) is the well-known log-sum inequality: for non-negative numbers and , denote the sum of all ’s by and the sum of all ’s by , and then we have
with equality if and only if are constant. Meanwhile, the discrete version of our result in (3) becomes that
Note that the extra term on the right side is instead of as we do not put any normalization conditions on or . 2) A comparison of (1) and (3) shows that we have an extra constant term for the second moment case, and thus it is natural to ask whether or not the term can be eliminated, i.e., whether it is always true that . The following counterexample provides a negative answer. Suppose that the takes three distinct values 0, 1, 2 with probabilities 29/36, 1/9, 1/12 under and with equal probabilities 1/3 under . Let be a function with a binary range {0,1} such that , . Then it is easy to verify that . More generally, other counterexamples can be easily found by choosing two distributions and of , both of which are supported on points such that the likelihood ratio and for with being distinct values. If we consider a quantization function that maps all to a single point but maps to another point , then , since the function is strictly concave on . In other words, unlike the case of Kullback-Leibler’s inequality (1), a quantization indeed can increase the second moment of the log-likelihood ratio. Fortunately, our theorem shows that such an increase is at most . However, it is unclear whether is sharp or not. 3) It is straightforward to generalize Proposition 1 to deal with the -moments of the absolute values of the log-likelihood ratios and for any real . Define and number . Then for any ,
higher-order moments of the log-likelihood ratios. For a positive integer , define the -th moment of the log-likelihood ratios as
That is,
, , where and are the log-likelihood ratios of and . Obviously, when is even, by the previous remark, we have , where is defined in (8). When is odd, it turns out that we need to define another constant to be the only real number that satisfies the equation
where is defined in (8). A simple numerical calculation shows that and . By applying the convex domination technique to the function for odd , we have
if the integer is odd. This includes the well-known Kullback-Leibler’s inequality (1) as a special case. 5) The convex domination technique can be used to derive a lower bound for the -th moments of log-likelihood ratios of both raw and quantized observations for all odd integer . By applying the convex domination technique to the function for odd integer , one can show that for an integer , the -th moments of log-likelihood ratios, and , are always no less than if is odd (and of course, are 0 if is even). For , this lower bound becomes 1 and thus the corresponding result is weaker than the well-known fact that and , the Kullback-Leibler divergences of raw and quantized observation, are always non-negative. ACKNOWLEDGMENT The authors are grateful to the associate editor and anonymous referees for their constructive comments that substantially improved an earlier version of this paper. REFERENCES
where the constant (8) and by convention that . 4) The convex domination technique we developed in proving Proposition 1 can be applied to deal with the
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Yan Wang received the B.S. degree in electronics from Peking University, Beijing, China, in 2001, the Ph.D. degree in mathematics from the University of Rome Tor Vergata, Roma, Italy, in 2007, and the Ph.D. degree in industrial engineering from the Georgia Institute of Technology, Atlanta, GA, USA, in 2011. Currently he is working at Pine River Capital Management, New York, NY, USA. His research interests include quickest detection, decentralized detection, and design of experiments.
Yajun Mei received the B.S. degree in mathematics from Peking University, Beijing, China, in 1996 and the Ph.D. degree in mathematics with a minor in Electrical Engineering from the California Institute of Technology, Pasadena, CA, USA, in 2003. He is currently an Associate Professor in The H. Milton Stewart School of Industrial and Systems Engineering at the Georgia Institute of Technology, Atlanta, GA, USA. He was a Postdoctoral Researcher in biostatistics and biomathematics in the Fred Hutchinson Cancer Research Center, Seattle, WA, USA, from 2003 to 2005, and was a New Research Fellow at the Statistical and Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park, NC, USA, during the fall semester of 2005. His research interests include quickest change detection, sensor networks, sequential analysis, and their applications in engineering and biomedical sciences.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 6, MARCH 15, 2013
Quantization Effect on the Log-Likelihood Ratio and Its Application to Decentralized Sequential Detection Yan Wang and Yajun Mei
Abstract—It is well known that quantization cannot increase the Kullback–Leibler divergence which can be thought of as the expected value or first moment of the log-likelihood ratio. In this paper, we investigate the quantization effects on the second moment of the log-likelihood ratio. It is shown via the convex domination technique that quantization may result in an increase in the case of the second moment, but the increase is bounded above by . The result is then applied to decentralized sequential detection problems not only to provide simpler sufficient conditions for asymptotic optimality theories in the simplest models, but also to shed new light on more complicated models. In addition, some brief remarks on other higher-order moments of the log-likelihood ratio are also provided.
is distributed according to for or 1. Then the Kullback-Leibler divergence of the quantized observation is
Index Terms—Convex domination, decentralized detection, Kullback-Leibler, log-sum inequality, quantization, quickest change detection, sequential detection.
(1)
I. INTRODUCTION
I
N information theory and statistics, the Kullback-Leibler divergence is a fundamental quantity that characterizes the difference between two probability distributions and of a random variable . Denote by and the densities of and with respect to some common underlying probability measure , then the Kullback-Leibler divergence from to is
where
is the log-likelihood ratio
and means taking the expectation over the distribution . In some applications, we need to deal with the quantized version of that is a function of and often is required to belong to a finite alphabet. Denote by the quantized observation, and let and be the probability distribution and probability mass (or density) function of when Manuscript received August 20, 2012; revised November 20, 2012; accepted November 27, 2012. Date of publication January 01, 2013; date of current version February 26, 2013. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Huaiyu Dai. This work was supported in part by the AFOSR grant FA9550-08-1-0376 and the NSF grants CCF-0830472 and DMS-0954704. The material in this paper was presented at the IEEE International Symposium on Information Theory, Cambridge, MA, USA, July 1–6, 2012. The authors are with the School of Industrial and System Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TSP.2012.2237170
where
is the log-likelihood ratio of
defined by
An important property is that quantization cannot increase the Kullback-Leibler divergence, that is,
with equality if and only if is a sufficient statistic of , see Theorem 4.1 of Kullback and Leibler [5]. This is conis generally less inforsistent with our intuition that mative than the itself. Note that the inequality (1), which will be referred to as Kullback-Leibler’s inequality below, deals with the expected value or first moment of the log-likelihood ratio. In this paper, we extend the Kullback-Leibler inequality (1) to investigate the quantization effects on the second or other higher moments of the log-likelihood ratio. Our research is motivated by the decentralized sequential detection problem where there ; ; are unobservable raw random variables , and where what is actually observed are their . In such a problem, one quantized versions ’s so as wants to design appropriate quantization functions ’s to make the best possible decision (under a to utilize the to maxproper criterion). Intuitively, it is natural to choose imize the Kullback-Leibler divergence of the quantized obser) if the ’s vations (or possibly or for each . See Tsitare independent with density siklis [18] for the characterization of such an optimal quantizer. In order to guarantee that the corresponding statistical procedures are indeed efficient, one needs to verify that the distrisatisfies some standard bution of quantized observations regularity conditions. One of such conditions is that quantized observations have finite variances, or more rigorously speaking, have that the second moments of the log-likelihood ratios a common upper bound over a class of allowable quantization ’s. functions Unfortunately, it can be analytically challenging or intractable to verify the assumptions for quantized observations ’s are known directly, even if the distributions of the to belong to some simple families of distributions, as one may to work with have a large class of quantization functions when finding the optimal one. To overcome such a difficulty, in this paper we show that one could dispense with quantization and only look at the second moment associated with the raw
1053-587X/$31.00 © 2012 IEEE
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data ’s since the boundedness of the unquantized version implies the boundedness of the quantized version regardless of the quantization function . Also see Le Cam and Yang [7] for similar approaches in other contexts. The remainder of the paper is organized as follows. In Section II we extend the Kullback-Leibler’s inequality (1) to the second moment of the log-likelihood ratio. In Section III, our result is applied to the decentralized sequential detection problems, not only to provide simpler sufficient conditions for asymptotic optimality theories in the simplest models, but also to shed new light on more complicated models. In Section IV, we add several remarks, including the quantization effects on higher-order moments of the log-likelihood ratios. II. SECOND-ORDER MOMENTS For the and , define their respective second moments of log-likelihood ratios as
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To do so, taking derivatives of the function leads to
Hence is convex on but is concave on . Therefore, if we consider the following new function if if
,
is a continuous convex function of when . then Moreover, the concavity of on implies that dominates . Now by the definitions of , and , we have
(2) where and are the log-likelihood ratios of and . Our main result of this paper is as follows. Proposition 1: Denote by the set of all possible measurable function . Then we have (3) Proof: Fix a measurable function , and let and be the likelihood ratios. To simplify notation, let denote the conditional expectation with respect to a given value of the quantization observation , then it is easy to see that
Recall that in the proof of the Kullback-Leibler’s inequality (1), one uses heavily the fact that the function is convex when . By Jensen’s inequality,
where the first inequality follows from the fact that , and the second inequality is an application of Jensen’s inequality to the new convex function . Meanwhile, the difference between and turns out to be insignificant. By definition, for all , and thus
where we use the fact that
Combining the above inequalities yields
for any given measurable function . Taking the supremum over all possible functions leads to (3), completing the proof of our theorem. and the Kullback-Leibler’s inequality (1) is proved by taking expectations under on both sides. Unfortunately, the above approach fails for the second moment case since the function is no longer convex (nor concave). Fortunately, this approach can be salvaged by what we call the “convex domination” technique, i.e., by finding a convex function that is larger, but not too much larger, than .
III. DECENTRALIZED SEQUENTIAL DETECTION The problem that motivated us to write the present paper arises from decentralized sequential detection problems. Below we will illustrate the usefulness of Proposition 1 to get around mathematical challenges in decentralized sequential detection problems, thereby providing new insights to the field.
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different formulations of these problems using different network architectures, we refer Braca et al. [2], [3] and the references herein. A. Decentralized Sequential Hypotheses Testing
Fig. 1. A decentralized sensor network.
As discussed in Bajovic et al. [1], distributed/decentralized detection can be broadly divided into several different classes, depending on parallel or consensus network architectures. Here we will only focus on a specific network system using the parallel architecture with a fusion center, see, Veeravalli, Basar and Poor [21], and Veeravalli [17], [19]. Fig. 1 depicts a typical configuration of such a network system. Suppose there are geographically deployed local sensors and a fusion center which makes a final decision. At each time step , each local sensor observes a raw data and sends a quantized message to the fusion center, which makes a final decision when stopping taking observations. Due to constraints on communication bandwidths or requirements of reliability, the local sensors are required to compress the raw data to quantized sensor messages ’s, which all belong to finite alphabets, say . In other words, the fusion center has no direct access to the raw observations and has to make its decisions based on the quantized sensor messages. If necessary, the fusion center can send feedback, , to the local sensors to adaptively adjust the local quantization so as to achieve maximum efficiency. There are many possible useful setups for the decentralized network system with the parallel architectures, and one widely used setup is the system with limited local memory and full feedback, or Case E in Veeravalli, Basar and Poor [21]. Mathematically, in such a system, the quantized sensor message sent from the local sensor to the fusion center at time is
where
(here ) denotes all past sensor messages at the fu-
sion center. In decentralized sequential detection problems, a fusion center policy often consists of selecting a stopping time at which time step it is decided to stop taking observations, and a final decision . The objective is to jointly optimize the sensor quantization functions ’s at the local sensor level and the fusion center policy so as to make an optimal decision in some suitable sense. Below we will discuss the usefulness of Proposition 1 in two different kinds of decentralized sequential detection problems: sequential hypothesis testing and quickest change detection. For
In the simplest model of decentralized (binary) sequential hypotheses testing problems, there are two simple hypotheses and regarding the distributions of the raw observations ’s, and conditional on each of these two hypotheses, the form i.i.d. sequences over time and are independent among different sensors. The objective is to use as few samples as possible to correctly decide which of these two simple hypotheses is true. That is, one wants to balances the trade-off between the average sample size under each hypothesis and the probabilities of making Type I and II errors. Under the Bays formulation, it is assumed that the two hypotheses and have known prior probability, say, for some , let be the cost of falsely rejecting , and each time step costs a positive amount . Then the Bayes risk of a decentralized sequential test with a stopping time is
where the notation is used to emphasize that the cost for each step is . The Bayes formulation of decentralized sequential hypothesis testing problems can be stated as follows. Problem (P1): Minimize the Bayes risk over all possible sensor quantization functions ’s and over all possible policies at the fusion center. The Bayes solution was characterized in Veeravalli, Basar and Poor [21], which showed that the Bayes solution is based on the “stationary Monotone Likelihood Ratio Quantizer (MLRQ)” quantizers. However, the optimal quantizers were “stationary” only in the sense of dynamic programming, i.e., the quantizers used thresholds to quantize observations, but the thresholds had to be updated at each time step, based on the posterior probability of . So, what was stationary was the functional structure of the quantizers. Specifically, for the Bayes procedure, the sensor messages of the -th sensor at time are formed through the following MLRQs based on the posterior probability :
where the
threshold functions satisfy
for all . While the threshold functions can be approximated via numerical experimentations, there is no closed-form formula for ’s, which appeared to be a discontinuous function of , see Fig. 3 and Fig. 4 of Veeravalli, Basar and Poor [21]. Thus, the usefulness of Bays procedure is rather limited, from both theoretical and practical viewpoint. In order to develop a feasible decentralized hypothesis testing procedure, it is natural to ask whether fixed quantizers can be used to construct decentralized sequential tests that are asymptotically optimal when the cost per time step goes
WANG AND MEI: QUANTIZATION EFFECT ON THE LOG-LIKELIHOOD RATIO
to 0. Unfortunately, the answer turns out to be negative, see Nguyen, Wainwright and Jordan [12] for a counterexample. The negative answer is not surprising, because each local sensor will have two kinds of optimal fixed quantizers: one maximizes (if is true) and the other maximizes (if is true), due to the asymmetric properties of the Kullback-Leibler divergences. Earlier Tsitsiklis [18] showed that the optimal quantizer that maximizes the local Kullback-Leibler divergence (or ) is a form of MLRQs. Denote them by and respectively. To develop a simple but asymptotically optimal decentralized sequential tests, Mei [10] introduces the concept of “tandem quantizers” where the test procedure is divided into two stages. In the first stage, any reasonable fixed quantizer is used and the network system makes a preliminary decision about which of two hypothesis is likely to be true. Then at the second stage, each local sensor switches to one of two optimal fixed quantizers or , based on whether the preliminary decision chooses the hypothesis or . The main motivation is that in statistics, first-order asymptotically optimal procedures often mainly depend on maximizing the first moment (or mean) of the log-likelihood ratios as long as the second moments of the log-likelihood ratios are finite. In particular, for the two-stage test with tandem quantizers, if the time steps taken at the first stage is large enough to make a reasonable preliminary decision, but relatively small as compared to those at the second stage, then we will asymptotically maximize the first moment of the log-likelihood ratios of quantized observations simultaneously under both and . Thus, one should expect that the two-stage test with tandem quantizers enjoy some nice properties. Indeed, it was shown in Mei [10] that under the sufficient condition
(4) for all , then the two-stage tests with tandem quantizers are first-order asymptotically optimal among all possible decentralized sequential tests that use any sequence of measurable quantizers, stationary or not. While the sufficient condition (4) has been verified in a couple of specific cases such as normal and exponential distributions with binary sensor messages, unfortunately it is unclear whether (4) holds in general or not. This paper is the result of our continued efforts to better understand the sufficient condition (4). By Proposition 1, the asymptotic optimality theory in Mei [10] holds under a simpler and more general sufficient condition, and can be stated as follows. Theorem 1: As long as the second moments of the log-likelihood ratios of raw sensor observations are finite, i.e., and for all , the two-stage tests with tandem quantizers are first-order asymptotically optimal among all possible decentralized sequential tests that use any sequence of measurable quantizers, stationary or not. We also want to add a remark to emphasize the important role of Proposition 1 in the more complicated model of decentralized sequential hypothesis testing problems in which one wants
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to test multiple hypotheses. Proposition 1 essentially guarantees that when developing asymptotically optimal decentralized sequential multi-hypothesis tests, one does not need to worry about the “strange” behavior of non-stationary quantizers (such as infinite variances), and it is paramount to choose local quantizers “smartly” so as to (asymptotically) maximize the first moment (or mean) of the log-likelihood ratios of quantized observations simultaneously under each and every possible hypothesis. This view allows us to extend the two-stage with tandem quantizers to develop asymptotic optimality theories for decentralized sequential multi-hypothesis testing problems, see [22] for the first-order asymptotic theorem and [23] for the second-order asymptotic theorem in certain cases. B. Decentralized Quickest Change Detection In decentralized quickest change detection problems, it is assumed that an event occurs to the network system at some unknown time , and changes the probability measure of the raw data from (with density for observations ) to (with density ). In the simplest model, it is assumed that all pre- and post-change densities, and , are completely specified, and conditional on each hypothesis on the possible change-time or (no change), the observations are independent over time and from sensor to sensor. Ideally, we want to continue taking observations as long as possible if no event occurs but will stop as soon as possible after the event occurs. A fusion center policy in decentralized quickest change detection problems can be simply defined as a stopping time , as its final decision at the stopping time is to claim that a change has occurred at or before time . There are two standard formulations to decentralized quickest change detection problems. The first one is the Bays formulation in which the change-time is assumed to be geometrically distributed, i.e., for , also see Shiryaev [15] and [16], Veeravalli [20]. Under the Bays formulation, it is further assumed that the cost of raising a false alarm is 1 and the cost of taking a post-change observation is per time step, and the problem can be stated as follows (For alternative minimax formulations, see Lai [6], Lorden [8], Moustakides [11], Page [13], Pollak [14], etc.) Problem (P2): Minimize the Bayes risk
over all possible fusion center stopping times and all possible sensor quantization functions ’s. The Bays procedure is characterized in Veeravalli [20], which again showed that the Bays solution is based on the MLRQ quantizers which were “stationary” only in the sense of the functional structure of the MLRQs, as the thresholds had to be updated at each time step, based on the posterior probability of change. Through numerical simulations, Veeravalli [20] found that as compared to Bays procedure, the loss in performance is insignificant if one simply uses fixed MLRQ quantizers whose thresholds are constant over time and optimized off-line to maximize the Kullback-Leibler divergence . This leads to a conjecture in Veeravalli [20] that the optimal
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fixed MLRQs are sufficient to construct asymptotically optimal decentralized change detection schemes over the class of all possible decentralized schemes that use any sequence of measurable quantizers, stationary or not. Unfortunately, it is not easy to prove or disprove Veeravalli’s asymptotic optimality conjecture, partly because the regularity conditions of the quantized observations are needed to do any reasonable asymptotic analysis. Indeed, all existing research has to put some restrictions on the sensor quantization functions, see Crow and Schwartz [4], Tartakovsky and Veeravalli [17]. A different approach was presented in Mei [9], which showed that Veeravalli’s conjecture holds under the following sufficient condition (5) is the second moment of log-likelihood ratio for quanwhere tized observations defined in (2). Proposition 1 significantly broadens the applicability of the asymptotic optimality theorem in Mei [9], and shows that Veeravalli’s conjecture holds under a simpler sufficient condition as stated in the following theorem. Theorem 2: As long as the second-moments of the log-likelihood ratios of the raw observations are finite, i.e.,
the optimal fixed MLRQs are sufficient to construct asymptotically optimal decentralized change detection schemes over the class of all possible decentralized schemes that use any sequence of measurable quantizers, stationary or not. Proof: By Lai [6], the asymptotic optimality conclusion holds under the following sufficient condition:
(6) where at time , i.e.,
and Here
is the probability measure when the change occurs is the likelihood ratio for the quantized data ,
with is probability mass function, i.e.,
.
By using Kolmogorov’s inequality for martingales, Mei [9] shows that (5) implies (6). By Proposition 1, if , then (5) holds and so does (6). Hence, the asymptotic optimality conclusion is proved. Besides simplifying the sufficient condition of asymptotic optimality theory in the simplest model when all pre- and
post-change densities, and , are completely specified for the local sensor observations, Proposition 1 can also shed new lights on a more complicated model of the decentralized quickest change detection problem when the post-change densities ’s are only partially specified. To the best of our knowledge, little research has been done due to its mathematical difficulties and complexities. To illustrate our main idea, let us focus on a concrete simplified example under a minimax formulation of decentralized quickest change detection problems. Assume there is only sensor, and the local raw sensor observations ’s are independent normally distributed with variance 1, with a possible change in the mean from 0 to one of two possible values, , for some known . The quantized sensor mesare binary, and the fusion center sages wants to utilize the ’s to raise an alarm quickly once a change occurs. Under the minimax formulation, when the true post-change mean is , one widely used definition of the detection delay is
where denotes the expectation of raw data when the mean change from 0 to at time . Now there are two kinds of detection delays: one for and the other for . A good decentralized quickest detection scheme should keep both detection delays small. Thus, our problem can be stated as follows. Problem (P3): Find a decentralized quickest detection scheme that nearly (or more rigorously, asymptotically) minimizes both and simultaneously over all possible fusion center stopping times and all possible sensor quantization functions ’s, subject to the false alarm constraint for some pre-specified constant . Here denotes the expectation when there are no changes. is a fixed quantizer over time , say, When for some threshold , the quantized sensor messages are independent Bernoulli random variable, and the fusion center essentially faces the problem of detecting a change on from to or . Such a problem is well-studied in the centralized quickest change detection problem, and one efficient scheme is the 2-CUSUM procedure defined by
where we use to emphasize that the 2-CUSUM procedure depends on the fixed quantizers in the decentralized quickest change problems. Here is the CUSUM statistic at time in the problem of detecting a change from to and is the CUSUM statistic at time in the problem of detecting a to . Specifically, for , the CUSUM change from statistic can be defined recursively as
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TABLE I THE DETECTION DELAYS OF FOUR PROCEDURES
with
, where and depend on the value of . It is natural to ask which is the “optimal” fixed quantizer , or what is the “optimal” fixed threshold for the 2-CUSUM procedure . It turns out the answer depends on what we mean by “optimal.” If we want to minimize , then we should choose to maximize , where and . On the other hand, if we want to minimize , then we should choose to maximize with . Moreover, it is not difficult to show that if we want to minimize , then we should choose to maximize . For instance, when , numerical computation shows that the corresponding three “op, 0.7941, and 0. timal” thresholds are A more challenging question is to develop a single decentralized change detection scheme that is asymptotically optimal and in the sense of asymptotically minimizing both simultaneously. In this case, the sensor message function cannot be fixed, and the thresh’s must be allowed to depend on the past messages olds . But how to choose the ’s? This is an open question so far, and our Proposition 1 provides some new ideas to tackle this problem. By Proposition 1, to develop asymptotically optimal decentralized change detection schemes, we can ignore the second moment and simply investigate how to choose local quantizers so as to (asymptotically) maximize the first moment (or mean) of the log-likelihood ratios of quantized observations simultaneously under all possible post-change hypotheses. Now we have two possible post-change means, and , and two corresponding optimal fixed thresholds, say for detecting and for detecting a a change in the mean is from 0 to change from 0 to . Thus the question can be further reduced to how to adaptively assign or to local sensor message functions so that it can “self-tune” to match the true post-change mean. Note that the idea of two-stage tests with tandem quantizers in decentralized sequential hypothesis testing problems no longer works for decentralize quickest change detection problems, since the change may happens after the first stage and thus the preliminary decision at the first stage does not necessarily reflect the true change. However, the fundamental
ideas can be salvaged: the local sensor message functions can be optimized according to a preliminary decision. To be more concrete, one possible strategy for decentralized quickest change detection can be defined as follows. Suppose that the fusion center uses the 2-CUSUM-type procedure of the form , with the detection statistic designed for monitoring a change from 0 to and the detection statistic for monitoring a change from . Initially one may use the quantizer 0 to at time . For , the quantizer used at time step will depend on the observed values of and . If , then we may feel that the more likely post-change mean will probably be and thus should be chosen as . On the other hand, if , we may think that the post-change mean is likely and thus . In particular, for , and . Now once we determine at time , the local sensor takes an the specific threshold observation , and sends the message to the fusion center, which uses relation (7) to update both and , except that the values of and depend on the specific value of . We call this procedure 2-CUSUM with adaptive thresholds, and denote it by . , we ran a numerical simulaTo see the advantage of tion for , and compared it with the 2-CUSUM procedure with three different fixed thresholds , 0.7941, 0. For these four procedures, the threshold value was first determined from the criterion , and we then use the value to simulate the detection delays and based on Monte Carlo repetitions. The results are reported in Table I. From Table I, all three 2-CUSUM procedures with fixed quantizers have some weakness: The threshold leads to a procedure that has smallest detection delay but the largest detection delay , and the conclusion . While the was reversed for the fixed threshold fixed threshold leads to a procedure that has similar performances on and , the detection delays are much larger than the smallest possible individual delays we can obtain. Meanwhile, the proposed scheme with adaptive thresholds ’s performs well under both post-change is close to its smallest scenarios in the sense that value for both and . All these are consistent with our theoretical conclusions.
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IV. REMARKS 1) The discrete version of the Kullback-Leibler’s inequality (1) is the well-known log-sum inequality: for non-negative numbers and , denote the sum of all ’s by and the sum of all ’s by , and then we have
with equality if and only if are constant. Meanwhile, the discrete version of our result in (3) becomes that
Note that the extra term on the right side is instead of as we do not put any normalization conditions on or . 2) A comparison of (1) and (3) shows that we have an extra constant term for the second moment case, and thus it is natural to ask whether or not the term can be eliminated, i.e., whether it is always true that . The following counterexample provides a negative answer. Suppose that the takes three distinct values 0, 1, 2 with probabilities 29/36, 1/9, 1/12 under and with equal probabilities 1/3 under . Let be a function with a binary range {0,1} such that , . Then it is easy to verify that . More generally, other counterexamples can be easily found by choosing two distributions and of , both of which are supported on points such that the likelihood ratio and for with being distinct values. If we consider a quantization function that maps all to a single point but maps to another point , then , since the function is strictly concave on . In other words, unlike the case of Kullback-Leibler’s inequality (1), a quantization indeed can increase the second moment of the log-likelihood ratio. Fortunately, our theorem shows that such an increase is at most . However, it is unclear whether is sharp or not. 3) It is straightforward to generalize Proposition 1 to deal with the -moments of the absolute values of the log-likelihood ratios and for any real . Define and number . Then for any ,
higher-order moments of the log-likelihood ratios. For a positive integer , define the -th moment of the log-likelihood ratios as
That is,
, , where and are the log-likelihood ratios of and . Obviously, when is even, by the previous remark, we have , where is defined in (8). When is odd, it turns out that we need to define another constant to be the only real number that satisfies the equation
where is defined in (8). A simple numerical calculation shows that and . By applying the convex domination technique to the function for odd , we have
if the integer is odd. This includes the well-known Kullback-Leibler’s inequality (1) as a special case. 5) The convex domination technique can be used to derive a lower bound for the -th moments of log-likelihood ratios of both raw and quantized observations for all odd integer . By applying the convex domination technique to the function for odd integer , one can show that for an integer , the -th moments of log-likelihood ratios, and , are always no less than if is odd (and of course, are 0 if is even). For , this lower bound becomes 1 and thus the corresponding result is weaker than the well-known fact that and , the Kullback-Leibler divergences of raw and quantized observation, are always non-negative. ACKNOWLEDGMENT The authors are grateful to the associate editor and anonymous referees for their constructive comments that substantially improved an earlier version of this paper. REFERENCES
where the constant (8) and by convention that . 4) The convex domination technique we developed in proving Proposition 1 can be applied to deal with the
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Yan Wang received the B.S. degree in electronics from Peking University, Beijing, China, in 2001, the Ph.D. degree in mathematics from the University of Rome Tor Vergata, Roma, Italy, in 2007, and the Ph.D. degree in industrial engineering from the Georgia Institute of Technology, Atlanta, GA, USA, in 2011. Currently he is working at Pine River Capital Management, New York, NY, USA. His research interests include quickest detection, decentralized detection, and design of experiments.
Yajun Mei received the B.S. degree in mathematics from Peking University, Beijing, China, in 1996 and the Ph.D. degree in mathematics with a minor in Electrical Engineering from the California Institute of Technology, Pasadena, CA, USA, in 2003. He is currently an Associate Professor in The H. Milton Stewart School of Industrial and Systems Engineering at the Georgia Institute of Technology, Atlanta, GA, USA. He was a Postdoctoral Researcher in biostatistics and biomathematics in the Fred Hutchinson Cancer Research Center, Seattle, WA, USA, from 2003 to 2005, and was a New Research Fellow at the Statistical and Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park, NC, USA, during the fall semester of 2005. His research interests include quickest change detection, sensor networks, sequential analysis, and their applications in engineering and biomedical sciences.
